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Finding 1-Dimensional substructures in set of kinematic time series in a cyclic motor task

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Recently using graph based data analysis is a popular approach when large datasets are collected and processed. In motor control this may be novel and here we apply it for a cyclic motor task. 18 ablebodied participants performed arm cranking exercises on an arm cycle ergometer (MEYRA, Germany). They rotated the crank of the ergometer with a cadence of 60 rpm. From each participant, data from twenty-four 30 seconds trials were collected by a ZEBRIS ultrasound based motion analyzer. The 3D coordinates of 8 markers were recorded (100Hz), 6 on the arm, 1 on the ergometer's handle and 1 reference marker. Surface EMGs were recorded (900Hz) from the biceps, triceps, delta anterior, delta posterior of both arms. For each trial, the time series of 24 coordinates (3 per markers from one arm) and 8 EMG time series gave the input to the graph based dimension reduction algorithm. We computed pairwise correlation coefficients of the 32 time series. Thus we got 32x31/2=496 numbers and defined a 32 by 32 symmetric matrix C. The i-th element of the j-th row of C is the correlation coefficient of the ith and j-th time series. Then we computed the determinants of all the 2 by 2 submatrices of C. We arranged the resulted values into a 496 by 496 symmetric matrix D. This gave the adjacency matrix of a graph G. If the entry in the i-th row and j-th column of D is close to 0, then nodes i and j are adjacent in G. A clique in this graph identifies a subset of the time series that can be substituted by one reference time series as they differ only in a scalar multiplier, thus they form a 1D structure. Our algorithm finds such 1D structures. We point out that principal component analysis may find the data set having a relatively high dimension. In the same time the new method may detect a number of 1-dimensional subsets in the data. A union of various 1-dimensional subsets can easily prevent the whole data set having an overall low dimension. We had large matrices and run the clique searching algorithm on G for all trials. We note that the computational time is strongly increasing with the number of time series. The most common clique, found in 96% of the trials, was a 4-clique. This corresponded to the time series (upward-downward movements) of 4 markers: 3 on the participant's hand and 1 on the ergometer's handle. These spatial points moved up-down in phase, in 96 % of the trials. For the forward-backward movements of these points this rate was 30%. In the lateral direction their movement weren't in phase 68 29th Annual NCM Meeting Toyama, Japan April 24 – 27, 2019 and didn't correspond to a clique. The latter was found for EMG time series of the studied muscles as well. Comparing the right and left arms the same results were found when identifying 1D subsets. These results were not surprising in arm cranking. Though, our algorithm identified those time series, which changed in phase and thus offered a research tool for studying other motor tasks and many time series, to find (maybe hybrid) kinematic and muscle synergies.